Chapter 6 Trend Forecasting Techniques
LEARNING OBJECTIVES
· 1. To examine the role of time intervals in forecasting models.
· 2. To examine guidelines for length of future forecasts.
· 3. To understand forecasting based on average change, average percent change, and confidence intervals.
· 4. To understand weighted forecasting models by using moving averages.
· 5. To understand weighted forecasting models by using exponential smoothing.
· 6. To be able to select between forecasting models based upon the calculated mean absolute deviation (MAD).
REAL WORLD SCENARIO
Molly Adel has determined that a mathematically derived forecasting model is preferable for her management needs. She is primarily interested in staffing and resource needs for the coming 12 months and has collected visit volume to Northern College Health Services by month for the past 60 months, shown in Appendix 6-A . After discussions with the admissions office, they do not predict any stark increases in applications or enrollment to the college; however, this is only a short-range projection, as the cost of other schools’ tuition, the availability of financial aid, and any necessary tuition increases will affect enrollment to the college, and thus demand for health services resources. She is also uncertain how useful data from 5 years ago are to projecting volume into the future as the student population has grown and services have changed in that time. Ms. Adel wants to be sure that her forecast is accurate enough to project resource needs into the future, but also flexible enough to anticipate changes in the external environment.
The methods that are described in this chapter are based upon the assumption that the past (and present) can predict the future. Some refer to these models and techniques as naive models in the sense that they only recognize and incorporate the past (and current) state into a forecast. They are naive as to the potential for small or major changes in the contexts that gave rise to the forecasts. In our real world example this might include a dramatic and unexpected enrollment into the college that results in an influx of students seeking care.
Being naive does not make these models inappropriate to use in forecasting. Usually, naive models are used when detailed information on the near past is available and the need is to forecast the near future. For these purposes they can be quite effective. Caution, however, must be taken when extending these forecasts too far into the future, or relying too heavily on past data. Careful consideration should also be given to any trends within historical data, such as seasonality factors, and adjustments made when forecasting ahead.
LEARNING OBJECTIVE 1: EXAMINE THE IMPORTANCE OF TIME INTERVALS IN FORECASTING
Time intervals define forecasts. It is essential that forecasts be prepared in the same time interval as the historical data. For example, if the historical data are expressed in weeks, the forecast should be expressed in weeks. If the historical data are expressed in years, the forecast should be expressed in years and not in months, weeks, or days. In contrast, if the historical data are expressed in days, it is acceptable to express a forecast in a larger time interval such as weeks, months, or years.
Assessing trends within time intervals is also essential. The primary goal of any forecast is to minimize the systematic error associated with the forecast. Some of this excess error can be minimized by critically examining the historical data and determining whether the time intervals are useful for future prediction. For example, if Ms. Adel would like to forecast the number of visits to the Urgent Care Clinic at Northern College Health Services for the month of September, she must determine if the number of visits that occurred in June, July, and August should be used in the forecast. Given that the student population in these summer months is very different than the student population in September, when many more students are back on campus, it may not be prudent to include only these months in her forecast.
In this example, previous Septembers are likely to have greater predictive power when forecasting the next September. In other words, June, July, and August are not equal to September in the sense that the phenomenon being forecasted (e.g., clinic visits) is fundamentally different in these summer months than in September, October, or November, when the campus is fully populated. Thus, the forecast is dependent upon a seasonal trend.
Using equal time intervals can also refer to the length of the time interval. Some months have a different number of weekdays and weekend days, and unequal number of days. In some situations, failure to recognize this may artificially distort the forecast.
Further, a day may not be a day. The number of clinic visits may (naturally) vary by day of the week. For example, the Urgent Care Clinic may be closed Saturdays and Sundays, or have shorter hours on weekend days than on a weekday. Utilization may be higher on certain days of the week, especially those days after a day the clinic was closed, such as a Monday. Although any forecast will have some degree of time interval distance, the challenge is to minimize the excess error in the forecast and achieve time intervals that are as equal as possible.
LEARNING OBJECTIVE 2: EXAMINE GUIDELINES FOR LENGTH OF FUTURE FORECAST
The first guiding principle of forecasting is to always plot the data. Doing so with the historical data provided by Ms. Adel would quickly reveal the seasonal downtrend that occurs during the months of June, July, and August.
Figure 6-1 Hospital Live Births for January through June Graph
Visually examining the data can show linear trends in the data, be they positive or negative. It can also provide clues to underlying trends and historical spikes or declines in the trend that can provide signal points to managers for investigation when possible. Perhaps the change was a one-time shock that could not be anticipated, or perhaps it is an event that reoccurs systematically over time, such as the review of reimbursement policy, legislative review, or administrative policy. When plotting the data, the x-axis is used to plot time and the y-axis is used to plot the variable of interest.
Second, the length of the forecast should generally not exceed one third the length of the historical data. If 24 months of historical data are being used, then the forecast should be no longer than 8 months, or one third of 24. If 6 weeks of historical data exist, the forecast should be no longer than 2 weeks into the future. This is a convention provided as a guide, not as a rule. It is important that the length of the forecast be appropriate given the historical data. Forecasting the next 5 to 10 years based upon 2 or 3 months of past data, regardless of the approach used, would be inappropriate.
Another guiding principle of forecasting is to be conservative. Being conservative requires an understanding of what is being forecasted, and how important the estimate is to the organization. It is important to know whether it is worse to be 10% or 20% high or 10% or 20% low with a forecast. For example, it is better to provide a high forecast for a hospital’s need for blood as a stock out condition is medically unacceptable and compromises the health of patients. Conversely, it is better to provide a low forecast for the number of hospital inpatient days because that forecast is used to establish budgets and it is usually easier to add temporary staff than reduce core staff. Forecasts should be sensitive to the positive and negative implications associated with being high or low from the actual. Some forecasting techniques permit the calculation of the standard deviation and thus confidence interval associated with the forecast. As stated in Chapter 2 , 1.96 standard deviations above and below the forecasted mean provides a 95% confidence interval of where the actual future value will fall. In some instances, setting the forecast to 1.96 standard deviations above the mean will provide a conservative forecast. Other times, setting the forecast to 1.96 standard deviations below the forecasted mean may be more conservative. Sometimes calling the forecast at the 50% level (forecasted mean) also is the conservative approach. In this chapter we will give examples of each. Again, knowing the implications of being high or low with a forecast is essential to an accurate and effective forecast.
A final guiding principle is that forecasts should be transparent. This means that all assumptions as well as calculations used in any forecasting technique should be clearly stated and replicable. If a manager provides a forecast that weights the data based more on current observations than historical ones, this, as well as the degree of weighting applied should be provided with the forecast. This is especially important when forecasting methods are reviewed by others, be they higher level managers or boards of directors, who will want to be able to make comments and/or approvals based on the methods used. Being transparent allows the manager to both solicit helpful input on forecasting techniques, but also ensures that if some unanticipated event should occur, the assumptions and process used to forecast were open and understood.
LEARNING OBJECTIVE 3: TO UNDERSTAND FORECASTING BASED ON AVERAGE CHANGE, AVERAGE PERCENT CHANGE, AND CONFIDENCE INTERVALS
Time series techniques identify a historical trend and base the forecast upon extending this trend into the future. At least three approaches can be used to do this: (1) Extrapolation based upon Average Change, (2) Extrapolation based upon a Confidence Interval, and (3) Extrapolation based upon Average Percent Change. For this section examine the data in Table 6-1 .
Before using any mathematical technique, the data must be plotted. Examining Figure 6-1 , the data plot or cloud of data suggests a linear relationship with a mostly positive slope. However, visits do drop somewhat in June. One method to extend or extrapolate this historical trend to forecast the number of births for July would be to use a ruler to draw a straight line that “best fit†the historical data plot. This would entail attempting to draw a line that is centered among the data points. Doing so, however, is prone to error unless the line is mathematically derived, which we explore in Chapter 7 . The methods described here, however, provide more systematic methods for forecasting these data.
Table 6-1 Hospital Live Births for January through June
Month
Number of births
January
77
February
81
March
83
April
85
May
87
June
85
Total
498
Average
83
If the data plot looks random, or like a circular cloud of data without any evident linear relationship, one should decompose the time series data. This means examining groupings of the data in pieces and then regrouping it. For example, multiple years of Januarys, multiple years of Februarys, and so forth can be examined to see if this method of plotting the data presents a different image of the data. Also try composing the data by adding months together to create quarters (3-month periods), or group data by days, weeks, months, or any other relevant time period to the organization or external environment. As manager you must attempt to construct the data in a manner that will best lend itself to forecasting.
Extrapolation Based upon Average Change
This approach to forecasting requires examining the month-to-month change that occurs in the data. Table 6-2 includes the month-to-month changes in the data and then computes the average of these changes. Note that month-to-month change is calculated in whole terms, and not the absolute value of the change. Once the average of the month-to-month change has been derived, a forecast of births for June can be prepared. The basis for this forecast is the mean or average level of births experienced over the history of the available data. This approach insures that no single value artificially distorts the forecast.
Using average change also requires that the midpoint of the data be identified. For this example, the midpoint of the data is 3.5, or the data point between the third and fourth data points. To determine the midpoint, a series of steps and a simple formula are used. For an odd number of data points, the midpoint is the number of data points (n) divided by 2 (n / 2). For an even number of data points, the formula becomes (n + 1)/2.
For this example, the midpoint of the data distribution is (6 + 1)/2 = 3.5. If the example had seven months of data, the midpoint would be 7 + 1 or 8 divided by 2 = 4.
Table 6-2 Month to Month Change in Hospital Live Births January through June
Month
Number of births
Change from previous month
January
77
February
81
4
March
83
2
April
85
2
May
87
2
June
85
−2
Total
498
 8
Ave
83
1.6
Md. Pt.
3.5
Forecast for July = 83 + (3.5 × 1.6) =
88.6
Extrapolation based on average change uses the following equation:
Forecast month (FM)
= Average of the data + (Midpoint × Average Change)
Equation 6-1
For this example, the average of the data = 83 births per month, the midpoint = 3.5, and the average change = 1.6 births.
Therefore our formula becomes:
FM
=
83 + (3.5 × 1.6)
=
83 + 5.6
=
88.6
Extrapolation Based upon Using Average Percent Change
Table 6-3 Percent Change in Hospital Live Births January through June
Month
Number of births
Change from previous month
% Change
Jan
77
—
—
Feb
81
 4
5.19%
Mar
83
 2
2.47%
Apr
85
 2
2.41%
May
87
 2
2.35%
Jun
85
−2
−2.30%
Sum
498
 8
10.13%
Ave
83
1.6
  2.03%
Forecast for July = 85 + (85 × .0203) =
 86.73
Extrapolation based on average percent change builds upon that based on average change by calculating the percent change in births from month to month. Table 6-3 revises the data to include the percent change in births from month to month. A common mistake in calculating the percentage change is juxtaposing the numerator and denominator. A rule of thumb when dealing with time series data is to remember to take the change between time periods (future − past/past). We can also take the change from month to month first (future − past) and then divide that by the most recent past time period.
Having determined that the average percent change per month is 2.03%, we use the formula of:
FM
= Most Recent Month + (Average Percent Change × Most Recent Month)
Equation 6-2
July forecast
=
June value + (2.03% of June value)
=
85 + (85 × 0.0203)
=
85 + 1.726
=
86.73 or 87 births rounded into real terms, as there is no such thing as 0.73 of a birth.
Extrapolation Based on a Confidence Interval
As the name implies, this method uses a confidence interval to forecast. It is important to remember that 1.96 standard deviations above and below the mean represents a 95% confidence interval. Revisit Chapter 2 for a more detailed discussion. A forecast based upon this method can be 95% confident that the actual number of future births will be included in this interval. The benefit to constructing a confidence interval forecast is that you can be more confident in your ability to accurately forecast the future period within a range. The caution is that as your confidence level increases, the accuracy with which you can predict lessens. This is the fundamental nature of confidence intervals in that as confidence increases, so does the size of the interval. It is possible to construct a 100% confidence interval, which is simply the entire range of possible values. As the interval narrows, the level of confidence drops.
Using confidence intervals as forecasts becomes helpful to the manager in a number of ways. One is when precision is not the primary motivation for forecasting. If, for example, the manager wishes to forecast visit volume for staffing purposes, knowing that at each level of staffing (e.g., adding one additional staffing unit) allows for moderate flexibility in volume to be handled. Thus a team of one physician’s assistant, one nurse, and one physician can see between 1 and 20 visits, but adding another physician’s assistant increases that volume to 35. If the manager forecasts using a confidence interval for the coming month and finds the interval to be 21 to 34 visits, she can accurately construct an adequate staffing plan. Another value confidence intervals provide is when they are analyzed within the context of the other forms of forecasting, which is explored later in the chapter.
To use confidence interval forecasting as an approach, the standard deviation and/or standard error is used. If the historical data represent the population of all data and not a sample, the standard deviation can be used. If, however, the historical data are only a sample of data, the standard error is used, which is simply the standard deviation divided by the square root of n. An example of a sample of historical data would be in attempting to develop a forecast of future births for all hospitals on a state based on only a sample of data from select hospitals. If the forecast is for just one hospital using that hospital’s data, the standard deviation will suffice. A more complete description of calculating the standard deviation as well as standard error appears in Chapter 2 .
Table 6-4 Extrapolation Based on a 95% Confidence Interval
Month
Number of births
Jan
77
Ave
83
Feb
81
Std. Dev.
3.27
Mar
83
95% C.I.
Apr
85
Upper = 83 + (1.96 × 3.27)
89.41
Lower = 83 − (1.96 × 3.27)
76.59
May
87
Jun
85
Sum
498
Forecast for July = (89 − 77) rounded with 95% level of confidence
Table 6-4 shows the calculation for extrapolation based on a 95% confidence interval. For example, note that one standard deviation = 3.27 births and that 1.96 standard deviations (the 95% Confidence Interval) = 1.96 × 3.27 = 6.41 births. For the example, the 95% confidence interval is the mean ± 1.96 standard deviations or 83 ± 6.41 or 83 + 6.41 = 89.41 births and 83 − 6.41 = 76.59 births. This provides a forecasted number of births for July based on a 95% level of confidence between 89.41 and 76.59 births or 89 and 77 births rounded into whole terms. The standard error could also be used if data were assumed to be from a sample.
Comparing Extrapolation Techniques
At this point it is important to compare the different forecasting methods for July births given the available data and using the different approaches ( Table 6-5 ).
Although each technique yields a different estimate, collectively the techniques provide sufficient information to venture a forecast. Choosing a technique is not an attempt to determine which provides the correct forecast for the next time frame. Each technique or method provides a “right†answer. These forecasts provide the manager with comparisons and options to select a forecast. Forecasting requires judgment, not just the ability to solve mathematical formula. That is, forecasting is an art as well as a science.
Table 6-5 Comparison of Extrapolation Techniques
Technique
July Forecast
Extrapolation based upon Average Change
87
Extrapolation based upon Average Percent Change
89
Extrapolation based upon 95% Confidence Interval
77–89
In retrospect, each of these basic methods is based upon different properties of the data used in the forecast. Each assumes a certain degree of linearity in the data and an inherent relationship between time (as the independent variable) and births (as the dependent variable) that has a positive or negative slope. For example, the level of births forecasted for July, in one instance 89, is higher than the actual number of births recorded for June (i.e., 85).
Although helpful in forecasting, these three techniques may be too simple for most applications. These techniques mask variability in the data. In the example used, some of this natural variability is based upon a different number of days per month. These methods also distill the data using the average or average percent change calculations. Except for the confidence interval approach, each method assumes that the forecast will be based on the theoretical line that best represents the past data. Therefore these methods are offered as a starting point for forecasting, not as definitive methods that can be relied upon exclusively to provide a relevant forecast.
LEARNING OBJECTIVE 4: TO UNDERSTAND FORECASTING BASED ON A MOVING AVERAGE
Because historical data often vary, sometimes considerably, it is often helpful to utilize a technique that does not tend to mask the inherent variability of the data, as do the methods described thus far. Moving averages correct for this and provide a method to examine the variability in data and use this pattern of variability in constructing a forecast.
To demonstrate moving averages the data in Table 6-1 have been revised by adding more historical data (July to December) and changing one of the original historical data points (March from 83 to 63). These data are restated in Table 6-6 .
Table 6-6 Hospital Live Births for January through June
Month
Number of births
July
68 
August
79 
September
81 
October
55 
November
71 
December
60 
January
77 
February
81 
March
63 
April
85 
May
87 
June
85 
Total
478 
Average
79.67
Figure 6-2 is the plot of the data included in Table 6-6 . Note that the scale used in this historical data plot has been selected to magnify the variability of the historical data.
Using a moving (time period to time period) average to forecast is a method that first examines the variability in the historical data and then provides the ability to mathematically “smooth or soften†the historical variability in search of a master or underlying trend. The first operation is to choose some time period over which to calculate the moving average. This we will term our n-time period. Here, n can represent a number of time periods, such as a 2-month (n = 2), 3-month (n = 3), 4-month (n = 4), or any n we choose given the availability of historical data. The steps to developing a moving average forecast are:
· 1. Select an n (n must be greater than 1). Because 12 months of historical data are included in Table 6-6 , n could be 2, 3, 4, 5, or 6. To begin, n = 2 has been selected.
· 2. Calculate the n-period moving average. To do this, start with the oldest data and work forward. For example, the forecast for September, using a 2-month moving average, is (July + August)/2. The forecast for October, using a 2-month moving average, is (August + September)/2.
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